A uniform thin wire is bent into a semicircle of radius r. Determine the coordinates of its center of mass with respect to an origin of coordinates at the center of the "full" circle.
Because wire is uniform x (sub cm) = 0
mass=dm
length=dl
coordinates (x,y) = (r cox theta, r sin theta)
To determine the coordinates of the center of mass of the semicircle, we need to calculate the x-coordinate and y-coordinate separately.
The x-coordinate of the center of mass, denoted as x_cm, can be calculated by integrating the product of the position (x) and mass elements (dm) over the entire length of the semicircle.
Since the wire is uniform, the mass is distributed uniformly along the length. Therefore, we can express mass element dm as a constant times the length element dl. Let's call this constant λ (lambda).
dm = λ * dl
Now, let's express the position element x in terms of an angle theta, where theta varies from 0 to π. The coordinates (x, y) of any point on the semicircle can be expressed as (r * cos(theta), r * sin(theta)), where r is the radius of the semicircle.
So, x = r * cos(theta)
To find the total length of the semicircle, we need to integrate the length element dl from 0 to π. The length element dl can be expressed as r * dθ, where dθ is the differential angle.
Now, let's set up the integral to calculate the x-coordinate of the center of mass:
x_cm = ∫ (x * dm) / M, where M is the total mass of the wire
Substituting the expressions for dm and x:
x_cm = ∫ (λ * r * cos(theta) * r * dθ) / M
Since the wire is uniform, we can assume λ * r and M are constants. We can move them outside the integral:
x_cm = (λ * r / M) * ∫ (cos(theta) * r * dθ)
The expression inside the integral is a straightforward integration:
x_cm = (λ * r / M) * ∫ (r * cos(theta) * dθ)
Evaluating the integral from 0 to π:
x_cm = (λ * r / M) * [r * sin(theta)] evaluated from 0 to π
Since sin(0) = 0 and sin(π) = 0, the x_cm simplifies to:
x_cm = 0
Therefore, the x-coordinate of the center of mass is 0. This means the center of mass lies on the y-axis.
Now, let's calculate the y-coordinate of the center of mass, denoted as y_cm. Similar to the x-coordinate calculation, we need to integrate the product of the position (y) and mass elements (dm) over the entire length of the semicircle.
The y-coordinate of any point on the semicircle is given by y = r * sin(theta).
Substituting y in the integral:
y_cm = ∫ (y * dm) / M
Again, substituting the expressions for y and dm:
y_cm = ∫ [(r * sin(theta)) * λ * r * dθ] / M
Moving the constants outside the integral and evaluating the integral from 0 to π:
y_cm = (λ * r^2 / M) * [-(cos(theta))] evaluated from 0 to π
Using the same logic as before, cos(0) = 1 and cos(π) = -1, the y_cm simplifies to:
y_cm = (λ * r^2 / M) * [-(1 - (-1))]
y_cm = (λ * r^2 / M) * (-2)
Since λ * r^2 is a constant, let's denote it as k:
y_cm = k * (-2) / M
Therefore, the y-coordinate of the center of mass is equal to -2k/M. The center of mass lies on the negative y-axis.
To summarize, the coordinates of the center of mass with respect to the origin at the center of the "full" circle are (0, -2k/M), where k is a constant determined by λ and r, and M is the total mass of the wire.